• Monte Carlo Simulation involves the use of pseudo random numbers to model systems where time plays no substantive role
(i.e., static models )
• Generation of artificial data through the use of a random number generator and the cumulative distribution of interest
• Random number generator
– Generates random variables that are uniformly distributed on the interval from 0 to 1 ( U (0, 1)) – Excel’s rand() function is an example
– Not possible to generate truly random numbers with a computer algorithm
• Random numbers, U (0, 1), are then transformed so that they follow the desired probability distribution.
– Uniform ( a , b )
– Normal ( µ , σ )
– Symmetric triangular ( a , b )
9/2/2003 Monte Carlo Simulation 2
• You are planning to invest a total of $15,000 and you have three investment vehicles from which to choose
• The return for each investment vehicle is a random variable ( R
L
R
M
, and R
H
, respectively) and the distribution for each of these random variables is given in the table below
,
• Use Monte Carlo simulation to characterize the distribution of the investment value at the end of one year based on a user-given allocation of the initial investment
9/2/2003
Investment Option
Low risk
Medium Risk
High Risk
Distribution of return (in %)
R
L
~ Normal (3, 1)
R
M
~ Normal (5, 5)
R
H
~ Normal (10, 15)
Monte Carlo Simulation 3
1

• The year-end value of the investment is given by the following expression:
V = S
L
(1 + R
L
) + S
M
(1 + R
M
) + S
H
(1 + R
H
)
9/2/2003 Monte Carlo Simulation 4
• Based on a model from Anthony Sun (used with permission)
– http://www.geocities.com/WallStreet/9245/vba12.htm
• A firm is considering producing and selling a new product under a pure/perfect competition market and the firm wants to know the probability distribution for the profit associated with this product.
• The total profit is given by the equation:
TP
(
) ( Q V F )
• Where:
– Q is the quantity sold
– V is the variable cost per unit
– P is the sales price per unit
– F is the fixed cost for producing the product
9/2/2003 Monte Carlo Simulation 5
• For this product, Q , P , and V are random variables with the following distributions:
– Q : uniform (80,000, 120,000)
– P : normal(22, 5)
– V : normal(12, 8)
• F is estimated to be 300,000
• Want to use Monte Carlo simulation to characterize the distribution of total profit for the proposed product
9/2/2003 Monte Carlo Simulation 6
2

• This problem is from Section 2.2 of Seila et al . (2003)
• A large catalog merchandiser is planning to have a special furniture promotion a year from now. To do this, the company must place its order for furniture now. It plans to sign a contract with the manufacturer for 3000 chairs at a cost of $175 per unit, which the company plans to offer initially for $250 per unit. The promotion will last for 8 weeks, after which all remaining units will be offered for sale at half the original price, or $125 per unit. The company believes that 2000 units will be sold during the first 8 weeks.
• While the number of chairs ordered and the ordering price are set contractually, the number chairs sold during the first 8 weeks and the initial selling price are actually random variables that depend on a number of environmental conditions.
• The company would like to characterize the distribution of the expected profit.
9/2/2003 Monte Carlo Simulation 7
• Assume the following distributions for the two stochastic inputs:
– Demand for chairs – symmetric triangular (500, 3500)
– Initial selling price – uniform (200, 300)
• The profit is given by:
P
(
)
R
2
)
• Use Monte Carlo simulation to characterize the expected profit for the furniture promotion
9/2/2003 Monte Carlo Simulation 8
3